On Mean Convex Regions

TL;DR

This paper proves that connected mean convex regions with multiple components cannot have positive mean curvature and provides decay estimates for mean curvature, extending results to hyperbolic space.

Contribution

It establishes a new geometric restriction on mean convex regions with multiple components and generalizes curvature decay estimates to hyperbolic space.

Findings

Connected mean convex regions with at least two components cannot have strictly positive mean curvature.

Derived decay estimates for mean curvature at infinity.

Extended curvature decay results to hyperbolic space.

Abstract

We prove that a connected mean convex region in $\mathbb{R}^{n+1}$ with at least two components cannot have strictly positive mean curvature. This answers a question of Gromov. We also obtain estimates for how quickly the mean curvature must decay at infinity, and generalize this result to hyperbolic space.